reserve A for QC-alphabet;
reserve n,k,m for Nat;
reserve F,G,G9,H,H9 for Element of QC-WFF(A);
reserve t, t9, t99 for Element of dom tree_of_subformulae(F);
reserve x for set;
reserve x,y for set;
reserve t for Element of dom tree_of_subformulae(F),
  s for Element of dom tree_of_subformulae(G);
reserve t for Element of dom tree_of_subformulae(F),
  s for FinSequence;
reserve C for Chain of dom tree_of_subformulae(F);
reserve G for Subformula of F;
reserve t, t9 for Entry_Point_in_Subformula_Tree of G;
reserve G1, G2 for Subformula of F,
  t1 for Entry_Point_in_Subformula_Tree of G1,
  s for Element of dom tree_of_subformulae(G1);

theorem Th37:
  s in G1-entry_points_in_subformula_tree_of G2 implies t1^s is
  Entry_Point_in_Subformula_Tree of G2
proof
  (tree_of_subformulae(F)).t1 = G1 by Def5;
  then
A1: t1 in F-entry_points_in_subformula_tree_of G1 by Def3;
  assume s in G1-entry_points_in_subformula_tree_of G2;
  then
A2: t1^s in F-entry_points_in_subformula_tree_of G2 by A1,Th27;
  F-entry_points_in_subformula_tree_of G2 c= dom tree_of_subformulae(F) by
TREES_1:def 11;
  then reconsider t9 = t1^s as Element of dom tree_of_subformulae(F) by A2;
  (tree_of_subformulae(F)).t9 = G2 by A2,Def3;
  hence thesis by Def5;
end;
